Typically, to guarantee sufficient exploration of arms, restrictions on the maximum (\(\pi _{\max }\) ) and minimum (\(\pi _{\min }\) ) possible values of the allocation probabilities are imposed [1]}, [2]}, [3]}. In such case, \(\pi _{t,k} \in [\pi _{\min },\pi _{\max }]\) , with \( 0 < \pi _{\min } \le \pi _{\max } < 1\)  and generally \(\pi _{\max } = 1- \pi _{\min }\) , for all \(t\) 's and \(k\) 's. In this work, we only need the algorithm to be randomized, i.e., \( 0 < \pi _{t,k} < 1\)  for all \(k\) 's and \(t \le T\). Note that this less restrictive condition holds for most popular MAB approaches, including TS.
